Some additional support for the apparently unreasonable conclusion that if it is Monday, it is more likely that the coin will land heads. Suppose that on each awakening, the experimenter flips a second coin, and if the second coin lands heads, the experimenter tells Beauty what day it is, and does not do so if it is tails.
If Beauty is told that it is Tuesday, this is evidence (conclusive in fact) that the first coin landed tails. So conservation of expected evidence means that if she is told that it is Monday, she should treat this as evidence that the first coin will land heads.
Some additional support for the apparently unreasonable conclusion that if it is Monday, it is more likely that the coin will land heads.
More likely than what?
Using per-awakening probabilities, ithe probability of heads without this information is 1⁄3.
The new information makes heads more likely than the 1⁄3 that the probability would be without the new information. It doesn’t make it more likely than 1⁄2.
I misplaced that comment. It was not a response to yours.
More likely than what?
More likely than .5. In fact I am saying the probability of getting heads is 2⁄3 after being told that it is Monday.
Using per-awakening probabilities, ithe probability of heads without this information is 1⁄3.
This is a frequentist definition of probability. I am using probability as a subjective degree of belief, where being almost certain that something is so means assigning a probability near 1, being almost certain that it is not means assigning a probability near 0, and being completely unsure means .5.
Here is how this works. If I am sleeping Beauty, on every awakening I am subjectively in the same condition. I am completely unsure whether the coin landed/will land heads or tails. So the probability of heads is .5, and the probability of tails is .5.
What is the subjective probability that it is Monday, and what is the subjective probability it is Tuesday? It is easier to understand if you consider the extreme form. Let’s say that if the coin lands tails, I will be woken up 1,000,000 times. I will be quite surprised if I am told that it is day #500,000, or any other easily definable number. So my degree of belief that it is day #500,000 has to be quite low. On the other hand, if I am told that it is the first day, that will be quite unsurprising. But it will be unsurprising mainly because there is a 50% chance that will be the only awakening anyway. This tells me that before I am told what day it is, my estimate of the probability that it is the first day is a tiny bit more than 50% -- 50% of this is from the possibility that the coin landed heads, and a tiny bit more from the possibility that it landed tails but it is still the first day.
When we transition to the non-extreme form, being Monday is still less surprising than being Tuesday. In fact, before being told anything, I estimate a chance of 75% that it is Monday -- 50% from the coin landing heads, and another 25% from the coin landing tails. And when I am told that it is in fact Monday, then I think there is a chance of 2⁄3, i.e. 50⁄75, that the coin will land heads.
This tells me that before I am told what day it is, my estimate of the probability that it is the first day is a tiny bit more than 50%… When we transition to the non-extreme form, being Monday is still less surprising than being Tuesday.
In the non-extreme form, the chance of being Monday is 2⁄3 and the chance of being Tuesday is 1⁄3. 2⁄3 is indeed less surprising than 1⁄3, so your reasoning is correct.
before being told anything, I estimate a chance of 75% that it is Monday -- 50% from the coin landing heads, and another 25% from the coin landing tails
Before being told anything, you should estimate a 2⁄3 chance that it’s Monday (not a 75% chance). There are three possibilities: heads/Monday, tails/Monday, and tails/Tuesday, all of which are equally likely. Because tails results in two awakenings, and you are calculating probability per awakening, that boosts the probability of tails, so it would be incorrect to put 50% on heads/Monday and 25% on tails/Monday. Tails/Monday is not half as likely as heads/Monday; it is equally likely. Only in the scenario where you were woken up either on Monday or Tuesday, but not both, would the probability of tails/Monday be 25%.
And when I am told that it is in fact Monday, then I think there is a chance of 2⁄3, i.e. 50⁄75, that the coin will land heads.
When you are told that it is Monday, the chance is not 50⁄75, it’s (1/3) / (2/3) = 50%. Being told that it is Monday does increase the probability that the result is heads; however, it increases it from 1⁄3 → 1⁄2, not from 1⁄2 → 2⁄3.
Before being told anything, you should estimate a 2⁄3 chance that it’s Monday (not a 75% chance). There are three possibilities: heads/Monday, tails/Monday, and tails/Tuesday, all of which are equally likely.
I disagree that these situations are equally likely. We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day. So obviously these two situations do not seem equally likely to me. And in particular, it seems equally likely to me that the coin was or will be heads, and that it was or will be tails. Going back to the non-extreme form, this directly implies that it seems half as likely to me that it is Monday and that the coin will be tails, as it is that it is Monday and that the coin will be heads. This results in my estimate of a 75% chance that it is Monday.
Because tails results in two awakenings, and you are calculating probability per awakening, that boosts the probability of tails, so it would be incorrect to put 50% on heads/Monday and 25% on tails/Monday. Tails/Monday is not half as likely as heads/Monday; it is equally likely.
I am not calculating “probability per awakening”, but calculating in the way indicated above, which does indeed make Tails/Monday half as likely as heads/Monday.
Only in the scenario where you were woken up either on Monday or Tuesday, but not both, would the probability of tails/Monday be 25%.
I am not asking about the probability that the situation as a whole will somewhere or other contain tails/Monday; this has a probability of 50%, just like the corresponding claim about heads/Monday. I am being asked in a concrete situation, “do you think it is Monday?” And I am less sure it is Monday if the coin is going to be tails, because in that situation I will not be able to distinguish my situation from Tuesday. And this is surely the case even when I am woken up both on Monday and Tuesday. It will just happen twice that I am less sure it is Monday.
And based on the above reasoning, being told that it is Monday does indeed lead me to expect that the coin will land heads, with a probability of 2⁄3.
We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day.
You should not be more surprised in that situation. The more days there are, the more that the extra tails awakenings push down the probability of heads. With 500000 awakenings, the probability gets pushed down by a lot. Now heads is 1⁄500001 per-awakening probability, same as tails-day-1 and tails-day-500000
You are claiming that if I will be wake up 500,000 times if the coin lands tails, I should be virtually certain a priori that the coin will land tails. I am not; I would not be surprised at all if it landed heads. In fact, as I have been saying, the setup does not make me expect tails in any way. So at the start the probability remains 50% heads, 50% tails.
I do not. I mean reporting my opinion when someone asks, “Do you think the coin landed, heads, or tails?” I will truthfully respond that I have no idea. The fact that I would be woken up multiple times if it landed tails, did not make it any harder for the coin to land heads.
I’d recommend distinguishing between the probability that the coin landed heads (which happens exactly once), and the probability that, if you were to plan to peak you would see heads (which would happen on average 250,000 times).
The problem is that you are counting frequencies, and I am not. It is true that if you run the experiment many times, my estimate will change, from the very moment that I know that the experiment will be run many times.
But if we are going to run the experiment only once, then even if I plan to peek, I would expect with 50% probability to see heads. That does not mean “per awakening” or any other method of counting. It means that if I saw heads, I would say, “Not surprising; that had a 50% chance of happening.” I would not say, “What an incredible coincidence!!!!”
Some additional support for the apparently unreasonable conclusion that if it is Monday, it is more likely that the coin will land heads. Suppose that on each awakening, the experimenter flips a second coin, and if the second coin lands heads, the experimenter tells Beauty what day it is, and does not do so if it is tails.
If Beauty is told that it is Tuesday, this is evidence (conclusive in fact) that the first coin landed tails. So conservation of expected evidence means that if she is told that it is Monday, she should treat this as evidence that the first coin will land heads.
More likely than what?
Using per-awakening probabilities, ithe probability of heads without this information is 1⁄3.
The new information makes heads more likely than the 1⁄3 that the probability would be without the new information. It doesn’t make it more likely than 1⁄2.
I misplaced that comment. It was not a response to yours.
More likely than .5. In fact I am saying the probability of getting heads is 2⁄3 after being told that it is Monday.
This is a frequentist definition of probability. I am using probability as a subjective degree of belief, where being almost certain that something is so means assigning a probability near 1, being almost certain that it is not means assigning a probability near 0, and being completely unsure means .5.
Here is how this works. If I am sleeping Beauty, on every awakening I am subjectively in the same condition. I am completely unsure whether the coin landed/will land heads or tails. So the probability of heads is .5, and the probability of tails is .5.
What is the subjective probability that it is Monday, and what is the subjective probability it is Tuesday? It is easier to understand if you consider the extreme form. Let’s say that if the coin lands tails, I will be woken up 1,000,000 times. I will be quite surprised if I am told that it is day #500,000, or any other easily definable number. So my degree of belief that it is day #500,000 has to be quite low. On the other hand, if I am told that it is the first day, that will be quite unsurprising. But it will be unsurprising mainly because there is a 50% chance that will be the only awakening anyway. This tells me that before I am told what day it is, my estimate of the probability that it is the first day is a tiny bit more than 50% -- 50% of this is from the possibility that the coin landed heads, and a tiny bit more from the possibility that it landed tails but it is still the first day.
When we transition to the non-extreme form, being Monday is still less surprising than being Tuesday. In fact, before being told anything, I estimate a chance of 75% that it is Monday -- 50% from the coin landing heads, and another 25% from the coin landing tails. And when I am told that it is in fact Monday, then I think there is a chance of 2⁄3, i.e. 50⁄75, that the coin will land heads.
In the non-extreme form, the chance of being Monday is 2⁄3 and the chance of being Tuesday is 1⁄3. 2⁄3 is indeed less surprising than 1⁄3, so your reasoning is correct.
Before being told anything, you should estimate a 2⁄3 chance that it’s Monday (not a 75% chance). There are three possibilities: heads/Monday, tails/Monday, and tails/Tuesday, all of which are equally likely. Because tails results in two awakenings, and you are calculating probability per awakening, that boosts the probability of tails, so it would be incorrect to put 50% on heads/Monday and 25% on tails/Monday. Tails/Monday is not half as likely as heads/Monday; it is equally likely. Only in the scenario where you were woken up either on Monday or Tuesday, but not both, would the probability of tails/Monday be 25%.
When you are told that it is Monday, the chance is not 50⁄75, it’s (1/3) / (2/3) = 50%. Being told that it is Monday does increase the probability that the result is heads; however, it increases it from 1⁄3 → 1⁄2, not from 1⁄2 → 2⁄3.
I disagree that these situations are equally likely. We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day. So obviously these two situations do not seem equally likely to me. And in particular, it seems equally likely to me that the coin was or will be heads, and that it was or will be tails. Going back to the non-extreme form, this directly implies that it seems half as likely to me that it is Monday and that the coin will be tails, as it is that it is Monday and that the coin will be heads. This results in my estimate of a 75% chance that it is Monday.
I am not calculating “probability per awakening”, but calculating in the way indicated above, which does indeed make Tails/Monday half as likely as heads/Monday.
I am not asking about the probability that the situation as a whole will somewhere or other contain tails/Monday; this has a probability of 50%, just like the corresponding claim about heads/Monday. I am being asked in a concrete situation, “do you think it is Monday?” And I am less sure it is Monday if the coin is going to be tails, because in that situation I will not be able to distinguish my situation from Tuesday. And this is surely the case even when I am woken up both on Monday and Tuesday. It will just happen twice that I am less sure it is Monday.
And based on the above reasoning, being told that it is Monday does indeed lead me to expect that the coin will land heads, with a probability of 2⁄3.
You should not be more surprised in that situation. The more days there are, the more that the extra tails awakenings push down the probability of heads. With 500000 awakenings, the probability gets pushed down by a lot. Now heads is 1⁄500001 per-awakening probability, same as tails-day-1 and tails-day-500000
You are claiming that if I will be wake up 500,000 times if the coin lands tails, I should be virtually certain a priori that the coin will land tails. I am not; I would not be surprised at all if it landed heads. In fact, as I have been saying, the setup does not make me expect tails in any way. So at the start the probability remains 50% heads, 50% tails.
Yes, I am (assuming you mean per-awakening certainty).
I do not. I mean reporting my opinion when someone asks, “Do you think the coin landed, heads, or tails?” I will truthfully respond that I have no idea. The fact that I would be woken up multiple times if it landed tails, did not make it any harder for the coin to land heads.
I’d recommend distinguishing between the probability that the coin landed heads (which happens exactly once), and the probability that, if you were to plan to peak you would see heads (which would happen on average 250,000 times).
The problem is that you are counting frequencies, and I am not. It is true that if you run the experiment many times, my estimate will change, from the very moment that I know that the experiment will be run many times.
But if we are going to run the experiment only once, then even if I plan to peek, I would expect with 50% probability to see heads. That does not mean “per awakening” or any other method of counting. It means that if I saw heads, I would say, “Not surprising; that had a 50% chance of happening.” I would not say, “What an incredible coincidence!!!!”